I address this more from the perspective of learning and less from the perspective of a display of knowledge. If the goal of university is to create good workers, then I believe open note/book is best, as it allows the student to approximate an environment in which they would be performing in an office. My qualm with this is – is this really what university should be for? It certainly has been during my MS and (short beginning of Ph.D) programs. If the student is coming in with the explicit goal of a college degree being to get them a higher paying job than would otherwise have been possible, I think that open book/note quizzes are best for the student. If the student comes in with the explicit goal of learning, regardless of the applicability of that learning to a job, then I believe that close book/note would be better.

At the same time, through COVID, I think that it is rather obvious that cheating has become (more?) rampant, as cheating is far easier on online courses than in person. I believe that open notes/book reduces this cheating enormously, and improves the validity and reliability of the quiz scores.

At the end of the day, I think that the question of being able to use material on a quiz/exam can only be answered in context of the purpose of the course. If the goal of the class is to establish a base of knowledge that the student will need as a foundation for other courses, I think that closed materials is best, as it forces the student to commit the information to memory, thus (hopefully) making this foundation firmer and easier to access in following courses. This is not the goal of all courses, and some courses, I believe, are made better by allowing the student to access materials, as it is unreasonable, or perhaps purposeless, to ask them to memorize certain materials.